Several properties related to interior and boundary of a set

We have the following propositions, which at the first glance seem to be true, but the truth still needs some careful consideration: If $U$ is open, $U=\Int(\overline{U})$. $\Bd(\Bd(A)) = Bd(A)$. $\Bd(\Int(A)) = \Bd(\overline{\Int(A)})$. For the first proposition, $\Int(\overline{U})$ represents the union of all the open sets which are contained in $\overline{U}$. $\because U \subset \overline{U}$ and $U$ is open, $\therefore U \subset \Int(\overline{U})$. If the inverse of this proposition is true, it implies that $U$ is the maximum among the open sets that are contained…

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Kuratowski's complement-closure theorem

Consider the power set $\mathcal{P}(X)$ of the topological space $X$. Given an arbitrary element in $\mathcal{P}(X)$, successive applications of the two operations: complement and closure can generate a series of subsets of $X$. The Kuratowski theorem states that the maximum number of distinct sets which can be generated from these operations is 14. Prove: At first, we need the definitions of interior and boundary of a set to separate the whole space $X$, which can be written as: \begin{equation} X = \Int(A) \cup \Bd(A) \cup…

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